We present O(log logn)round algorithms in the Massively Parallel
Computation (MPC) model, with ˜O(n) memory per machine, that
compute a maximal independent set, a 1 + ε approximation of
maximum matching, and a 2 + ε approximation of minimum vertex
cover, for any nvertex graph and any constant ε > 0. These improve
the state of the art as follows:
• Our MIS algorithm leads to a simple O(log log Δ)round
MIS algorithm in the CONGESTEDCLIQUE model of distributed
computing, which improves on the ˜O (plog Δ)round
algorithm of Ghaffari [PODC’17].
• OurO(log logn)round (1+ε)approximate maximum matching
algorithm simplifies or improves on the following prior
work: O(log2 logn)round (1 + ε)approximation algorithm
of Czumaj et al. [STOC’18] and O(log logn)round (1 + ε)
approximation algorithm of Assadi et al. [arXiv’17].
• Our O(log logn)round (2+ε)approximate minimum vertex
cover algorithm improves on an O(log logn)round O(1)
approximation of Assadi et al. [arXiv’17].
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Massively Parallel Algorithms for Distance Approximation and Spanners.
Over the past decade, there has been increasing interest in distributed/parallel algorithms for processing largescale graphs. By now, we have quite fast algorithmsusually sublogarithmictime and often poly(łogłog n)time, or even fasterfor a number of fundamental graph problems in the massively parallel computation (MPC) model. This model is a widelyadopted theoretical abstraction of MapReduce style settings, where a number of machines communicate in an alltoall manner to process largescale data. Contributing to this line of work on MPC graph algorithms, we present poly(łog k) ε poly(łogłog n) round MPC algorithms for computing O(k^1+o(1) )spanners in the strongly sublinear regime of local memory. To the best of our knowledge, these are the first sublogarithmictime MPC algorithms for spanner construction.
As primary applications of our spanners, we get two important implications, as follows: For the MPC setting, we get an O(łog^2łog n)round algorithm for O(łog^1+o(1) n) approximation of all pairs shortest paths (APSP) in the nearlinear regime of local memory. To the best of our knowledge, this is the first sublogarithmictime MPC algorithm for distance approximations. Our result above also extends to the Congested Clique model of distributed computing, with the same round complexity and approximation guarantee. This gives the first sublogarithmic algorithm for approximating APSP in weighted graphs in the Congested Clique model.
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 Award ID(s):
 2006664
 NSFPAR ID:
 10339759
 Date Published:
 Journal Name:
 33rd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2021)
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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